2002 Dodge Neon Car won't start in cold weather

Was just recently given the car. It worked fine until it started to turn to winter.

For some reason the battery would die once the weather started to drop around the freezing mark. Boosting it worked and would run just fine. If the car was stored in the garage, it would hold the charge longer than it would if it was sitting outside in the cold but would eventually lose its charge (I am assuming its because the battery was old and would not hold its charge anyways).

I decided to go buy a new battery from Canadian Tire, a Motormaster 72. I put the battery in the car and it started up perfectly. Put the car in the garage and the next day went to start it which it does with no problems. I brought the Neon up to my girlfriends work to swap vehicles because I needed our other car. It only sat in the parking lot for an hour and when my girlfriend went to start her car it wouldn't start.

So there is a brand new battery that starts in a garage no problem but once in the elements (cold weather) for only an hour it loses its charge.

Why does a brand new battery work it the garage but once put outside for only an hour, it will lose its charge. What could be the problem with this car? Is it possible that I bought a bad battery?